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Time Series

Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce

Complete topic list for Time Series in Maharashtra Board Class 12 Mathematics & Statistics -Commerce. Key concepts, sub-topics, and what to focus on for board exams.

44 questions5 concepts

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4 Topics · Maharashtra Board Class 12 Mathematics & Statistics -Commerce

Topics in Time Series

1

Definite Integral as Limit of Sum

  • Definite integral ∫ᵇₐ f(x) dx represents the area under curve y = f(x) between x = a and x = b
  • Derived from dividing interval [a,b] into n equal parts and taking limit as n → ∞
  • Formula: ∫ᵇₐ f(x) dx = lim(n→∞) Σᵣ₌₁ⁿ h·f(a + rh) where h = (b-a)/n
2

Fundamental Theorem of Integral Calculus

  • If F(x) is antiderivative of f(x), then ∫ᵇₐ f(x) dx = F(b) - F(a)
  • Connects differentiation and integration as inverse operations
  • Eliminates need for limit calculations in most practical problems
3

Properties of Definite Integrals - Basic Properties

  • Property I: ∫ₐₐ f(x) dx = 0 (integral over zero width interval)
  • Property II: ∫ᵇₐ f(x) dx = -∫ₐᵇ f(x) dx (swapping limits changes sign)
  • Property III: ∫ᵇₐ f(x) dx = ∫ᵇₐ f(t) dt (integration variable doesn't matter)
4

Advanced Properties - Substitution Properties

  • Property V: ∫ᵇₐ f(x) dx = ∫ᵇₐ f(a + b - x) dx (substitution property)
  • Property VI: ∫₀ₐ f(x) dx = ∫₀ₐ f(a - x) dx (special case for [0,a])
  • Property VII: ∫₀²ₐ f(x) dx = ∫₀ₐ f(x) dx + ∫₀ₐ f(2a - x) dx (interval doubling)

Key Concepts

The definite integral ∫ᵇₐ f(x)dxIf F(x) is an antiderivativeEight key properties simplify definite integralFor even functions f(For integrals of the form ∫₀^π/2

Frequently Asked Questions

What are the important topics in Time Series for Maharashtra Board Class 12 Mathematics & Statistics -Commerce?
Time Series covers several key topics that are frequently asked in Maharashtra Board Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Time Series — Maharashtra Board Class 12 Mathematics & Statistics -Commerce?
Understand the core concepts first, then work through the 44 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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